Densitometer with reduced sensitivity to pressure

ABSTRACT

A densitometer in the present disclosure comprises a piston attached to an end of a tube of the densitometer to reduce pressure dependence of density estimates of a sample fluid. The densitometer measures sample fluid density by vibrating the tube containing sample fluid and measuring the resonant frequency of the tube, then estimating the sample fluid density based on this resonant frequency. The piston is designed with a predetermined diameter that converts pressure inside the tube to tension in the tube. This tension produces an opposite effect on the resonant frequency of the tube to that caused by the fluid pressure itself and thereby reduces pressure dependence of the sample fluid density estimates.

BACKGROUND

The disclosure generally relates to the field of logging while drilling(LWD), and to pumpout wireline formation testing.

In wireline and LWD operations, accurate fluid density measurements areimportant for formation sampling and fluid identification. An in-linedensitometer can be deployed in a pumpout wireline formation tester(PWFT) for fluid density measurements. Using an in-line densitometer tomeasure fluid density is sensitive to changes in temperature andpressure downhole, as accuracy of the measurement by the in-linedensitometer relies on a characterization of the mechanical response ofthe densitometer under varying operating conditions includingtemperature, pressure, external forces, etc.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the disclosure may be better understood by referencing theaccompanying drawings.

FIG. 1 is a schematic diagram of a densitometer with dissimilar tube andclamp materials.

FIG. 2 is a schematic diagram of a densitometer with tension measuringdevices.

FIG. 3 is a schematic diagram of a densitometer with decreased pressuredependence.

FIG. 4 is an isometric view of a sealed piston at the end of adensitometer tube.

FIG. 5 is a frontal view of four strain gauges affixed to a tube in“axial full Poisson bridge” configuration.

FIG. 6 is a flowchart of example operations for designing andcalibrating a densitometer with minimized sensitivity to temperaturechanges.

FIG. 7 is a flowchart of example operations for designing andcalibrating a densitometer system with tension-measuring devices.

FIG. 8 is a flowchart of example operations for designing andcalibrating a densitometer system with minimized sensitivity topressure.

FIG. 9 depicts an example computer system with a densitometer fluiddensity estimator.

FIG. 10 is a schematic diagram of a drilling rig system with adensitometer.

FIG. 11 depicts a schematic diagram of a wireline system with adensitometer.

FIG. 12 depicts a graph of resonant frequency of a sample fluid in adensitometer versus temperature for a standard densitometer design anddifferent clamp coefficient of thermal expansion values.

FIG. 13 depicts a graph of axial force on the vibrating tube versustemperature for a standard densitometer design and different clampcoefficient of thermal expansion values.

FIG. 14 depicts a graph 1400 of resonant frequency versus pressure for astandard densitometer design.

FIG. 15 depicts a graph of resonant frequency versus pressure for astandard design densitometer without a tension-producing piston tocompensate for pressure.

FIG. 16 depicts a graph of resonant frequency versus pressure for apressure-compensated densitometer with an optimized piston diameter.

FIG. 17 depicts a graph of net axial force on the tube versus pressurefor the densitometer design in FIG. 16.

FIG. 18 depicts a graph of resonant frequency versus pressure for arevised pressure compensated densitometer.

FIG. 19 depicts a graph of net axial force on the tube versus pressurefor the densitometer design in FIG. 18.

DESCRIPTION

The description that follows includes example systems, methods,techniques, and program flows that embody aspects of the disclosure.However, it is understood that this disclosure may be practiced withoutthese specific details. For instance, this disclosure refers to wirelineformation testing in illustrative examples. Aspects of this disclosurecan be instead applied to production or injection well testing. In otherinstances, well-known instruction instances, protocols, structures andtechniques have not been shown in detail in order not to obfuscate thedescription.

Overview

A single-tube densitometer measure fluid density using a vibrationsource that drives a sample fluid cavity to resonance and measures aresonant frequency of the sample fluid. The fluid density measurement isrecovered as a function of the measured resonant frequency of the samplefluid. The densitometer receives fluid in stream during LWD operations,or wireline pumpout operations, via a tube which redirects all or someof the fluid into a vibrating tube section supported at each end by arigid body (clamp). The resonant frequency of the tube section betweenthe two clamp ends and containing the fluid sample is a function of boththe actual fluid density in the tube, as well as several other physicalcharacteristics of the densitometer and its environment, such as thetemperatures of the clamp and tube, the coefficients of thermalexpansion (CTE) of the clamp and tube, the axial pre-tension forceretained in the tube between the clamps, any external force exerted onthe tube outside of the clamp, the Young's moduli of the tube and clamp,the inner and outer diameter of the tube, the inner and outer diameterof the clamp segment between the clamp ends, the density of the tubematerial, the length of the tube segment between the clamp ends, thepressure of the fluid, etc. Errors in the fluid density measurement comefrom measurement errors of various sensors deployed throughout thedensitometer and also depend on an accurate theoretical characterizationof the resonant frequency with respect to various sensor measurements(e.g. the fluid pressure and temperature, the tube and clamptemperatures). A measurement module that determines the resonantfrequency as a function of these sensor measurements is calibrated basedon this theoretical characterization as well as predetermined systemdependencies on pressure, temperature, etc. Thus, errors in the fluiddensity measurement come from measurement errors of various sensorsdeployed throughout the densitometer or if the theoreticalcharacterization of the resonant frequency with respect to the varioussensor measurements is inaccurate.

Accuracy in fluid density measurement is influenced by accuratecalibration of a measurement module for a single-tube densitometer. Toincrease accuracy in calibration and correspondingly fluid densitymeasurement, the single-tube densitometer is designed to reduce thetemperature sensitivity of the densitometer. Using materials withdissimilar coefficient of thermal expansions for the tube and clampreduces temperature sensitivity. Moreover, using dissimilar clamp andtube materials allows for cost effective design considerations. Anaccurate theoretical characterization of the densitometer should accountfor clamp material and tube material with different CTEs. With thereduced temperature sensitivity facilitated by the clamp and tubematerials with different CTEs, the measurement module is more accuratelycalibrated due to reduced accumulation of errors from temperaturechanges and inaccurate temperature measurements, and the clamp materialand tube material can be chosen as a cost-efficient metallic alloy. Themeasured resonant frequency and fluid density derived thereof achievegreater accuracy when compared to a densitometer with identical tubematerial and clamp material.

To accurately calibrate the measurement module (and, therefore,accurately measure the fluid density), it is additionally desirable toreduce the dependency of the resonant frequency on the fluid pressure.Another improvement described in this disclosure targets the reductionof the pressure sensitivity of the densitometer. This is accomplished bychanging the hydraulic configuration of the fluid flow within thedensitometer to use fluid pressure to convey axial tension on the tube.The effect of the added axial tension caused by pressure on the tuberesonant frequency is equal and opposite to the usual effect due topressure and this added axial tension reduces or eliminates the effectof fluid pressure on the measured resonance frequency. This, in turn,reduces or eliminates the effect of pressure on the estimated density ofthe sample fluid by the measurement module.

It is found in practice with real densitometers that some externalforces applied to the tube are unpredictable. Those include thefrictional forces due to O-rings present in the densitometer assembly.These O-rings are needed as seals between the densitometer tube and theother tool components in which the densitometer is mounted. Themagnitude and direction of O-ring friction between the tube and thestructural parts holding the densitometer to the tool are, in general,indeterminate and this adds an unknown error to the estimated density ofthe sample fluid. Another improvement described in this disclosure isthe addition and use of tension measuring devices situated on the tubemeasure the axial tensile or compressive force the tube experiencesduring operational conditions. The measurement module is calibrated touse these measurements to provide a more accurate estimation of samplefluid density.

Example Densitometer with Minimized Temperature Dependence

FIG. 1 is a schematic diagram of a densitometer with dissimilar tube andclamp materials. A fluid 100 enters the densitometer through a tube 101and flows horizontally from left to right. A clamp 103 is attached tothe tube 101 and is contained in a housing 131 that shields the tube 101and the clamp 103 from external environmental factors such as forces,external fluids and pressure. A vibration source 105 and a vibrationdetector 106 are affixed to the tube 101. Typically, the vibrationsource 105 and the vibration detector 106 are magnets whose motion areactuated or detected by electromagnetic coils placed in the clamp (notpictured). A material-compensated fluid density estimator 102 iscommunicatively coupled to the vibration source 105 and the vibrationdetector 106 as well as various pressure, temperature, strain and forcesensors throughout the densitometer 190-192. Although depicted as 3sensors, more or less than 3 sensors can be implemented at variouslocations throughout the densiometer. The tube 101 has an inner diameterb 117, an outer diameter α 115, and an inner tube section inside theclamp of length L 121. The clamp 103 has an inner diameter B 111 and anouter diameter A 119. The clamp 103 and the tube 101 are made ofdifferent materials as indicated by different fill patterns in FIG. 1.The housing 131 can be made of material that is different than both theclamp 103 and the tube 101 or made of the same materials as one of thetube 101 and the clamp 103. The tube 101 is made of a material havingYoung's modulus E_(c), coefficient of thermal expansion (CTE) α_(c), andtemperature T_(c) as indicated by material properties 123. The clamp 103is made of a material having Young's modulus E_(t), CTE α_(t), andtemperature T_(t) as indicated by material properties 125. Typicaldimensions are b=0.219″, a=0.301″, L=6″, B=0.6″, A=1.8″, total tubelength 15″ (including sections outside the clamps) and clamp length 9″(clamp length not pictured). The section of tube of length L containedwithin the clamp is typically given pre-tension, held by the clamp. Atypical pre-tension value is +700 pound-force (lbf). However, thedensitometer may also be operated with the tube free of tension, or incompression (negative force value).

The vibration source 105 can comprise a magnet affixed to the outersection of the tube 101 and one or more electromagnetic coils affixed tothe inner section of the clamp 103. The material-compensated fluiddensity estimator 102 can drive an alternating current through theelectromagnetic coils which produces an oscillating force on the magnetaffixed to the tube 101, vibrating the section of tube 101 of length Lbetween the clamp ends. The vibration detector 106 can also comprise amagnet affixed to the tube 101 and one or more electromagnetic coilsaffixed to the inner section of the clamp 103. Vibrations in the tube101 cause the magnet in the vibration detector 106 to vibrate(vertically, in the plane of FIG. 1) which induces an alternatingcurrent in the electromagnetic coils in the vibration detector 106. Thevibration detector 106 sends current induced by the varying magneticfield to the material-compensated fluid density estimator 102 which canmeasure the current and infer a measured resonant frequency of the fluid100 inside the tube 101. Although depicted on the same side of the tube101, the vibration source 105 and the vibration detector 106 can beaffixed to opposite sides of the tube 101. The position of the vibrationsource 105 and the vibration detector 106 is designed to maximize theeffectiveness of vibrations in the tube 101 induced by the vibrationsource 105 and to minimize interference of magnetic fields created bythe vibration source 105 on vibration detector 106. Other configurationsof magnets and electromagnetic coils can be implemented and other typesof vibration sources and vibration detectors can be used that induce avibration in the tube 101 and accurately measure the resonant frequencyof the vibration. For example, vibration source 105 and vibrationdetector 106 can be part of a resonant electrical circuit designed tomaintain the vibrating tube section at resonance. In another embodiment,an optical fiber sensor is bonded to the tube and interrogated fordynamic strain.

A partial differential equation (PDE) can be derived for thedensitometer in FIG. 1 using first principles that describes thetransverse displacement of the tube 101, Ψ(x,t), at position x and timet as a result of the vibration:

$\begin{matrix}{{{E_{t}I\frac{\partial^{4}\psi}{\partial z^{4}}} + {\left( {{m_{f}V^{2}} - F_{t} + {P \cdot S_{f}}} \right)\frac{\partial^{2}\psi}{\partial z^{2}}} + {2m_{f}V\frac{\partial^{2}\psi}{{\partial t}\;{\partial z}}} + {\left\lbrack {\left( {m_{t}m_{f}} \right) + {M_{1}{\delta\left( {z - z_{1}} \right)}} + {M_{2}{\delta\left( {z - z_{2}} \right)}}} \right\rbrack\frac{\partial^{2}\psi}{\partial t^{2}}}} = 0} & (1)\end{matrix}$Here,

$I = {\frac{\pi}{64}\left( {a^{4} - b^{4}} \right)}$is the area moment of inertia of the tube 101,

$S_{f} = {\frac{\pi}{4}b^{2}}$is the cross-sectional area of the fluid 100 inside the tube 101,m_(f)=ρ_(f)S_(f) is the linear density (mass per unit length) of thefluid 100 inside the tube 101 (where ρ_(f) is the fluid density of thefluid 100), F_(t) is the horizontal axial force acting on the tube 101(in the vibrating section of length L), V is the average velocity of thefluid 100 inside the tube 101, P is the pressure of the fluid 100 insidethe tube 101, m_(t) is the linear density of the tube 101, M₁ is themass of a source magnet which is at position z₁, M₂ is the mass of thedetector magnet which is at position z₂, and δ(z) is the Dirac deltafunction. By neglecting the m_(f) V² term in (1) (with the reasoningthat removing this term will have a negligible effect on the solution ofthe PDE), (1) can be solved analytically, and the solution Ψ will beperiodic with time. The fundamental resonant frequency of the vibrationcan be derived by solving for the period of the solution Ψ in the t(time) variable. The resulting resonant frequency is:

$\begin{matrix}{f_{0} = {\frac{\beta_{0}^{2}}{2\pi\; L^{2}}\sqrt{\frac{E_{t}I}{m_{t} + {\rho_{f}S_{f}}}}}} & (2)\end{matrix}$Here, β₀=β₀(ρ_(f), T_(t), T_(h), P, E_(t), I, M₁, M₂, E_(t), α, b, A, B,z₁, z₂, F_(t)) is a function on all of the physical parameters of thedensitometer, where T_(t) is the temperature in the tube 101 and T_(h)is the temperature in the housing 131. Further, certain of theseparameters are dependent on the pressure and temperature of variouscomponents of the densitometer. For example, α=α(T_(t),P) andb=b(T_(t),P) are both dependent on the temperature and pressure in thetube 101 and E_(t)=E_(t)(T_(t)) is a function of the temperature of thetube 101. The material-compensated fluid density estimator 102, whenderiving the resonant frequency of the fluid 100 using equation (2), canbe calibrated to tune parameters that are dependent onpressure/temperature using pressure and temperature measurementsthroughout the densitometer and known (possibly pre-calibrated)dependence of these parameters.

To compensate for the use of different materials in the tube 101 and theclamp 103, the length, inner diameter, outer diameter, and force actingon the tube 101 can be approximated using physical properties of thetube 101 and clamp 103 materials. In the below equations (again derivedfrom first principles of the densitometer system), it is assumed that atension F_(t) ₀ was applied at the ends of the tube 101 prior totightening the clamp 103 and that this occurred while the tube 101 andthe clamp 103 were both at a reference temperature T_(ref) (e.g., roomtemperature). L₀, α₀, b₀ are the original values for L, a, b used inequation (1) at T_(t)=T_(ref), ν_(t) is Poisson's ratio for the materialof the tube 101, F_(Δ) is any external axial forces on the tube 101,

$S_{c} = {\frac{\pi}{4}\left( {A^{2} - B^{2}} \right)}$is the cross-sectional area of the clamp 103, and

$S_{t} = {\frac{\pi}{4}\left( {a^{2} - b^{2}} \right)}$is the cross-sectional area of the tube 101. Using all of theseparameters, F_(t), L, α, and b can be computed in equation (2) using thefollowing formulae:

$\begin{matrix}{F_{t} = {{\frac{E_{c}S_{c}}{{E_{c}S_{c}} + {E_{t}S_{t}}}\left\lbrack {\left( {F_{t_{0}} + \frac{\pi\;{Pv}_{t}b^{2}}{2}} \right) + {E_{t}{S_{t}\left( {{\alpha_{c}\left( {T_{c} - T_{ref}} \right)} - {\alpha_{t}\left( {T_{t} - T_{ref}} \right)}} \right)}}} \right\rbrack} + \frac{E_{t}S_{t}F_{\Delta}}{{E_{c}S_{c}} + {E_{t}S_{t}}}}} & (3) \\{L = {L_{0}\left\lbrack {1 - \frac{\pi\;{Pv}_{t}b_{0}^{2}}{2E_{t}S_{t}} + {\alpha_{t}\left( {T_{t} - T_{ref}} \right)} + \frac{F_{t}}{E_{t}S_{t}}} \right\rbrack}} & (4) \\{a = {a_{0}\left\lbrack {1 + \frac{\pi\;{Pb}_{0}^{2}}{2E_{t}S_{t}} + {\alpha_{t}\left( {T_{t} - T_{ref}} \right)} - \frac{F_{t}v_{t}}{E_{t}S_{t}}} \right\rbrack}} & (5) \\{b = {b_{0}\left\lbrack {1 + {\frac{P}{E_{t}}\left( {\frac{\pi\left( {a_{0}^{2} + b_{0}^{2}} \right.}{4S_{t}} + v_{t}} \right)} + {\alpha_{t}\left( {T_{t} - T_{ref}} \right)} - \frac{F_{t}v_{t}}{E_{t}S_{t}}} \right\rbrack}} & (6)\end{matrix}$The

$\frac{E_{c}S_{c}}{{E_{c}S_{c}} + {E_{t}S_{t}}}$factor in equation (3) compensates forces on the tube 101 for complianceof the clamp 103. For instance, for pre-tensioning the tube, an axialforce F_(t0) is applied before the clamps are tightened. Once the clampis tightened, the tension is released for the tube section outside theclamp. However, for the section of the tube 101 inside the clamp 103,the contribution to force F_(t) of this pre-tension is reduced due tothe fact that the clamp 103 is made with a material of finite Young'smodulus. If, after the initial release of tension, a new force F_(A)(tensile or compressive) is applied on the section of the tube 101outside of the clamp 103, this new force will affect the tensionremaining in the tube in the section inside the clamp 103. However, theeffect of the external force is going to be much smaller in the sectionof tube 101 inside the clamp 103. The reduction factor is

$\frac{E_{t}S_{t}}{{E_{c}S_{c}} + {E_{t}S_{t}}}.$This factor is typically of the order of 0.05.Using the values given in equations (3), (4), (5), and (6) in equation(2), the material-compensated fluid density estimator 102 can solve forthe resonant frequency of the tube for a given fluid 100 when the tube101 and clamp 103 are made of different materials, and also compensatefor temperature/pressure dependency of various parameters throughoutequations (2), (3), (4), (5), and (6). Moreover, the density ρ_(f) ofthe fluid 100 can be solved algebraically from equation (2) using thederived resonant frequency.

Although hidden by the catch-all term β₀ in equation (2), the resonantfrequency of the fluid 100 depends on the axial force along the tube 101F_(t) such that an increased axial force causes a higher resonancefrequency. Moreover, from equation (3) it is clear that the axial forceF_(t) is itself dependent on uniform temperature change in both tube 101and clamp 103 temperature T_(t), and T_(c) respectively. This can becharacterized by

$\begin{matrix}{\frac{\Delta\; F_{t}}{\Delta\; T} = {\frac{E_{c}S_{c}E_{t}S_{t}}{{E_{c}S_{c}} + {E_{t}S_{t}}}\left( {\alpha_{c} - \alpha_{t}} \right)}} & (7)\end{matrix}$From equation (7), if the tube 101 and clamp 103 are made from the samematerial (α_(c)=α_(T)), the temperature dependence of the axial force onthe tube vanishes. However, it is often cost-effective due to the designof the tube 101 and the clamp 103 to use dissimilar materials. Forexample, it may be cost-effective to use Titanium Ti-6Al-4V Grade 5Alloy for the material of the tube 101 and INCONEL® 706 PrecipitationHardening Alloy for the material of the clamp 103. The materials for thetube 101 and the clamp 103 can be chosen using equation (7) such that

$\frac{\Delta\; F_{t}}{\Delta\; T}$compensates for shift of the resonant frequency with a temperaturechange ΔT uniform over both the clamp and the tube. This will have theeffect of minimizing the dependence on uniform temperature changes ofthe resonant frequency and resultant fluid density derived from equation(2). In some embodiments, the axial load on the tube 101 using materialswith CTE prescribed by equation (7) will be beyond the threshold axialforce limits of the components of the densitometer. For example, usingthe above titanium-alloy tube and clamp may only be able to handle 25kilopounds per square inch (ksi) whereas using H.C. Starck® MP35NNickel/Cobalt/Chromium/Molybdenum and the above combination of MP35Nalloy and INCONEL alloy may only be able to handle 30 ksi. These choicesof materials reduce the temperature dependence of the resonant frequencyof the tube 101 instead of completely cancelling it out. Choosingmaterials with CTEs that completely cancel dependence may exceedacceptable load thresholds (measured in ksi above) on the tube 101 andclamp 103.

Because of the limited choice of specific coefficient of thermalexpansion values for materials practical to use for the tube 101 andclamp 103 in a downhole densitometer, the full cancellation of thetemperature dependence may not be possible with the choice of a singlepair of materials. Instead, the tube 101 and/or clamp 103 can be made ofa composite of cost-effective materials having various CTE values inorder to achieve a desired CTE. For example, the clamp 103 can comprisemultiple layers of a low CTE material (e.g., a graphite/epoxy composite)and layers of a high CTE material (e.g., thin aluminum layers). Otherlayer/material combinations are possible. Given a tube 101 and/or clamp103 comprising N layers, where the ith layer has a CTE of α_(i), Young'smodulus E_(i), and cross-sectional area S_(i), the effective CTE of thelayered materials is given by

$\alpha_{c,{eff}} = \frac{\sum\limits_{i = 1}^{N}\;{\alpha_{i}E_{i}S_{i}}}{\sum\limits_{i = 1}^{N}\;{E_{i}S_{i}}}$and this value can be used in all of the equations (2), (3), (4), (5),(6), and (7) for α_(c).Example Densitometer with Tension Measuring Devices

While desirable for the external axial force F_(Δ) to be zero to have noinfluence on the densitometer response, the connections of the tube in atypical tool with the other sections of the wireline tool or the LWDtool will be such that this term will depend on the fluid pressure. Forexample, the tube end can be connected to a manifold assembly (notpictured) with O-rings providing a seal between the tube 101 and themanifold. An increase of fluid pressure would then cause an increase ofthe axial compressive force acting at the tube end. Whereas thecontribution of fluid pressure on F_(Δ) can be calculated given ameasurement of the fluid pressure, the frictional force due to thesliding of the O-rings relative to the tube receptacle is typicallyunknown. This is because friction depends on the load history. Inparticular, friction reverses direction when the relative displacementbetween the parts reverses direction and causes hysteresis. We describehere how calibration and operation of the densitometer can utilizetension measuring devices attached to at least one of the tube and theclamp to account for the F_(Δ) term, including the frictional effectsthat would otherwise not be quantifiable.

FIG. 2 is a schematic diagram of a densitometer with tension measuringdevices. The role of the tension measuring devices is to providemeasurements from which the axial force applied to the tube outside ofthe clamp can be derived. With a direct measurement of this force, therole played by the mounting hardware, friction from O-rings, the effectof fluid pressure on the tube end, etc., can be taken into account inthe calculation of density, therefore providing a better estimate ofdensity. Referring to FIG. 2, fluid 200 enters the densitometer througha tube 201 and flows horizontally from left to right. A clamp 203 isattached to the tube 201 and is contained in a housing 231 that shieldsthe tube 201 and the clamp 203 from external environmental factors suchas forces, external fluids, pressure, etc. Tension measuring devices 233and 235 adhere to opposite sides of the tube 201 and measure strains ε₁,ε₃ 241 and ε₂, ε₄ 243, respectively. Tension measuring devices 237 and239 adhere to opposite sides of the clamp 203 and measure strains ε₅, ε₇245 and ε₆, ε₈ 247, respectively. A vibration source 205 and a vibrationdetector 206 are affixed to the tube 201. A tension-compensated fluiddensity estimator 202 is communicatively coupled to the vibration source205, the vibration detector 206, the tension measuring devices 233, 235,237 and 239, as well as various pressure, and temperature, strain andforce sensors throughout the densitometer (not pictured). The tube 201has an inner diameter b 217, an outer diameter α 215, and an inner tubesection length L 221, and the clamp has an inner diameter B 211, anouter diameter A 219. The housing 231, the clamp 203, and the tube 201can have different materials as indicated by different fill patterns inFIG. 2. The tension-compensated fluid density estimator 202 can beconfigured to compensate for different materials of the tube 201 asdescribed variously with reference to FIG. 1. Although in thisdescription we refer to strain gauges mounted on the tube, strain gaugesmounted on the clamp, and other force measurement devices such as loadcells which can be mounted at different locations along the densitometercan also be used. Not all of them are needed to determine the axialforce. For example, strain gauges mounted on the tube only may besufficient. Alternatively, strain gauges mounted on the clamp only maybe sufficient. However, strain changes due to external force on the tubeare much smaller on the clamp compared to the tube so the accuracy andprecision of the force determination will be poorer using strain gaugeson the clamp compared to using strain gauges on the tube.

The tension measuring devices 233, 235, 237, and 239 can be sets ofstrain gauges or any other device configured to measure strain on thetube 201 and the clamp 203. The measured strains 241, 243, 245, and 247comprise multiple measurements for each tension measuring device. Insome embodiments, a single strain can be measured at each tensionmeasuring device. The tension measuring devices 233, 235, 237, and 239are depicted as measuring two strains which can correspond to one straingauge aligned parallel to the axis of the tube 201 and one strain gaugealigned perpendicular to the axis of the tube 201. Such a configurationcan accurately measure axial strain along the tube 201 and is depictedwith reference to FIG. 5. For this embodiment, where ε₁ and ε₂ aremeasured by strain gauges arranged parallel to the axis of the tube 201and ε₃ and ε₄ are measured by strain gauges arranged perpendicular tothe axis of the tube 201. These strains are responsive to axial tensionon the tube, as well as to the pressure of the fluid inside the tube. Toestimate the axial tension, while minimizing the effect of temperatureon the strain gauges on the tube are connected in a Wheatstone bridgeconfiguration that outputs the following “bridge output” strain:ε_(B)=¼(ε₁−ε₂−ε₃−ε₄)  (8)where the ε_(i) values, with i=1,2,3 or 4, designate ratios ofresistance change to original resistance values (i.e., ΔR_(i)/R_(i)).

In order to accurately calibrate the densitometer in FIG. 2 whenderiving the resonant frequency from equation (2), the external forceF_(Δ) on the tube 201 appearing in equation (3) needs to be accuratelydetermined. The measured strain ε_(B) from equation (8) can be used toaccurately determine any external forces acting on the tube 201. In thefollowing derivations, the measured strains ε₁ and ε₃ are assumed to beequal and the measured strains ε₂ and ε₄, for the purposes of extractingan analytical formula for the external force F_(Δ) as a function ofε_(T). These measured strains may not be equal in practice, due tobending, whence equation (8) uses averages. Using standard strain gaugeformulae,

$\begin{matrix}{ɛ_{1} = {\frac{\Delta\; R_{1}}{R_{1}} = {ɛ_{3} = {\frac{\Delta\; R_{3}}{R_{3}} = {{K_{1}\left( {ɛ_{z} - {\alpha_{g}\Delta\; T}} \right)} + {K_{2}\left( {ɛ_{\theta} - {\alpha_{g}\Delta\; T}} \right)} + {K_{3}\Delta\; T}}}}}} & (9) \\{ɛ_{2} = {\frac{\Delta\; R_{2}}{R_{2}} = {ɛ_{4} = {\frac{\Delta\; R_{4}}{R_{4}} = {{K_{1}\left( {ɛ_{\theta} - {\alpha_{g}\Delta\; T}} \right)} + {K_{2}\left( {ɛ_{z} - {\alpha_{g}\Delta\; T}} \right)} + {K_{3}\Delta\; T}}}}}} & (10)\end{matrix}$where ε_(z) is the strain on the tube 101 in the axial direction, ε_(θ)is the strain on the tube 101 in the perpendicular direction (i.e. thedirection of the strain gauges perpendicular to the tube 101), α_(g) isthe CTE of the strain gauge material, ΔT is a uniform temperature changeacross the tube 201 and the clamp 203, and K₁, K₂, and K₃ are determinedusing parameters provided by the strain gauge manufacturer.Specifically, where the strain gauge manufacturer provides a gaugefactor G, a transverse sensitivity factor K_(t), and a Poisson ratio ofa reference sample ν₀, then

${K_{1} = \frac{G}{1 - {v_{0}K_{t}}}},$and K₂=K_(t)K₁ (K₃ does not need to be calculated as in cancels out inlater computations). The terms ε_(z) and ε_(θ) can be expressed as afunction of F_(Δ) (which will enable solving for F_(Δ) as a function ofε_(T)) by decomposition into their pressure, force, and temperaturedependencies:

$\begin{matrix}{ɛ_{z} = {{ɛ_{z}^{P} + ɛ_{z}^{F_{\Delta}} + ɛ_{z}^{\Delta\; T}} = {\frac{{- 2}{Pv}_{t}b^{2}}{E_{t}\left( {a^{2} - b^{2}} \right)} + \frac{F_{\Delta}}{E_{t}\frac{\pi}{4}\left( {a^{2} - b^{2}} \right)} + {\alpha_{T}\Delta\; T}}}} & (11) \\{ɛ_{\theta} = {{ɛ_{\theta}^{P} + ɛ_{\theta}^{F_{\Delta}} + ɛ_{\theta}^{\Delta\; T}} = {\frac{2{Pb}^{2}}{E_{t}\left( {a^{2} - b^{2}} \right)} + \frac{{- v_{t}}F_{\Delta}}{E_{t}\frac{\pi}{4}\left( {a^{2} - b^{2}} \right)} + {\alpha_{T}\Delta\; T}}}} & (12)\end{matrix}$plugging (11) and (12) into (9) and (10) and solving for ε_(B) usingequation (8),

$\begin{matrix}{ɛ_{B} = {\frac{\left( {K_{1} - K_{2}} \right)\left( {1 - v_{t}} \right)}{E_{t}\left( {a^{2} - b^{2}} \right)}\left\lbrack {{Pb}^{2} + {\frac{2}{\pi}F_{\Delta}}} \right\rbrack}} & (13)\end{matrix}$Solving for F_(Δ), we get:

$\begin{matrix}{F_{\Delta} = {\left( {\frac{ɛ_{B}{E_{t}\left( {a^{2} - b^{2}} \right)}}{\left( {K_{1} - K_{2}} \right)\left( {1 - v_{t}} \right)} - {Pb}^{2}} \right)\frac{\pi}{2}}} & (14)\end{matrix}$The tension-compensated fluid density estimator 202 is configured tocompute F_(Δ) using equation (13) with the corresponding measurementsand strain gauge parameters, and to use this value in equation (3) whencomputing the resonant frequency via equation (2).

Although depicted outside of the clamp 203, the tube-mounted tensionmeasuring devices 233 and 235 can be situated inside the clamp 203. Inthis case, the tube tension F_(t) can be obtained directly from thebridge output as follows:

$\begin{matrix}{F_{t} = {\left( {\frac{ɛ_{B}{E_{t}\left( {a^{2} - b^{2}} \right)}}{\left( {K_{1} - K_{2}} \right)\left( {1 - v_{t}} \right)} - {Pb}^{2}} \right)\frac{\pi}{2}}} & (15)\end{matrix}$If the four strain gauges are mounted on the clamp, then

$\begin{matrix}{ɛ_{1} = {\frac{\Delta\; R_{1}}{R_{1}} = {ɛ_{3} = {\frac{\Delta\; R_{3}}{R_{3}} = {{K_{1}\left( {ɛ_{z,c} - {\alpha_{g}\Delta\; T}} \right)} + {K_{2}\left( {ɛ_{\theta,c} - {\alpha_{g}\Delta\; T}} \right)} + {K_{3}\Delta\; T}}}}}} & (16) \\{ɛ_{2} = {\frac{\Delta\; R_{2}}{R_{2}} = {ɛ_{4} = {\frac{\Delta\; R_{4}}{R_{4}} = {{K_{1}\left( {ɛ_{\theta,c} - {\alpha_{g}\Delta\; T}} \right)} + {K_{2}\left( {ɛ_{z,c} - {\alpha_{g}\Delta\; T}} \right)} + {K_{3}\Delta\; T}}}}}} & (17) \\{ɛ_{z,c} = {\frac{F_{\Delta}}{{E_{t}S_{t}} + {E_{c}S_{c}}} + {\alpha_{c}\Delta\; T}}} & (18) \\{ɛ_{\theta,c} = {{{- v_{c}}\frac{F_{\Delta}}{{E_{t}S_{t}} + {E_{c}S_{c}}}} + {\alpha_{c}\Delta\; T}}} & (19) \\{ɛ_{B} = {{\frac{\left( {K_{1} - K_{2}} \right)}{2}\left( {ɛ_{z,c} - ɛ_{\theta,c}} \right)} = {{\frac{\left( {K_{1} - K_{2}} \right)\left( {1 - v_{c}} \right)}{2}\frac{F_{\Delta}}{{E_{t}S_{t}} + {E_{c}S_{c}}}} = {\frac{(G)\left( {1 - K_{T}} \right)\left( {1 + v_{c}} \right)}{2\left( {1 - {v_{0}K_{T}}} \right)}\frac{F_{\Delta}}{{E_{t}S_{t}} + {E_{c}S_{c}}}}}}} & (20)\end{matrix}$The force is obtained from:

$\begin{matrix}{F_{\Delta} = {\frac{2{ɛ_{B}\left( {{E_{t}S_{t}} + {E_{c}S_{c}}} \right)}}{\left( {K_{1} - K_{2}} \right)\left( {1 + v_{t}} \right)} = \frac{2{ɛ_{B}\left( {1 - {v_{0}K_{T}}} \right)}\left( {{E_{t}S_{t}} + {E_{c}S_{c}}} \right)}{{G\left( {1 - K_{T}} \right)}\left( {1 + v_{t}} \right)}}} & (21)\end{matrix}$Example Densitometer with Minimized Pressure Dependence

As discussed above, fluid pressure directly affects the resonancefrequency of the densitometer and this effect is separate from thechange of density of the fluid that occurs due to the compressibility ofthe fluid. All fluids are compressible to some extent and therefore willchange density with changes in pressure. The effects being consideredfor increased accuracy of the densitometer are the effects on theresonance frequency of the pressure itself, separate from the change ofdensity. Equation 1 indicates that pressure should have an individualimpact on the densitometer response, separate from the fluid densityρ_(f). In addition, pressure will have an effect on the dimensions ofthe tube, and on the axial force F_(t) in the tube, as made clear inEquations 3 to 6. Calibration of the densitometer can make use of theeffect of fluid pressure on the tube section outside of the clamp totailor the response of the densitometer to changes in fluid pressure.This reduces or eliminates the intrinsic sensitivity to pressure of thedensitometer, allowing for increased accuracy of measurements by thedensitometer.

FIG. 3 is a schematic diagram of a densitometer with decreased pressuredependence. Fluid 300 enters the densitometer through a tube 301 andfollows a flow path from left to right. A clamp 303 is attached to thetube 301 and is contained in a housing 327 that shields the tube 301 andthe clamp 303 from external environmental factors such as forces,external fluids, pressure, etc. A vibration source 305 and a vibrationdetector 306 are affixed to the tube 301. A pressure-compensated fluiddensity estimator 302 is communicatively coupled to the vibration source305, the vibration detector 306, as well as various pressure,temperature, strain and force sensors throughout the densitometer (notpictured). The tube 301 has an inner diameter b 317, an outer diameter α315, and an inner tube section length L 321. The clamp has an innerdiameter B 311 and an outer diameter A 319. The densitometer furthercomprises a first manifold inflow section 390 where the fluid 300originates at pressure P 340, and a tube section 392 where the fluidundergoes vibrations. The fluid 300 flows into the tube via port 331.The fluid 300 exerts pressure 340 on an annular surface 356 of a piston333 having a diameter c 343 as well as the tube end disk, which is partof the same piston, as indicated by the three arrows with label “P” inFIG. 3. The piston 333 is sealed to manifold 337 via O-rings 345/349(these two numbers refer to the same O-ring on opposite side of the tubein the cross-sectional drawing) and 347/351 (also a single O-ring). Themanifold 337 is also sealed to the tube 301 via the pair of O-rings353/357 and, 355/359. Although 4 total O-rings are depicted, the numberof O-rings can vary. A fluid chamber 335 situated behind the piston 333collects possible fluid leakage from the seal with the O-rings 345, 347,349, and 351. The housing 327, the clamp 303, and the tube 301 can havedifferent materials as indicated by different fill patterns in FIG. 3and the pressure-compensated fluid density estimator 302 can beconfigured to also compensate for the thermal response due to differentmaterials as described variously with reference to FIG. 1.

The dimension of the outer diameter of piston 333 relative to otherdimensions (including the diameter at O-rings 353/357 and 355/359) isadjusted during the design of the densitometer in FIG. 3 such that theresonant frequency of the vibrating section 392 of the tube 301 oflength L 321 has a minimized dependence on the pressure P. The keyadjustment desired is the amount of tension applied to the tube 301 inresponse to the applied pressure. In some embodiments, one of the setsof O rings (either the ones of piston 333, one the opposite ones shownas 353/357 and 355/359, can be replaced with a flexible disk or membranethat allows the relative displacement of the tube relative to themanifold 337 with applied pressure. Using a flexible disk or membraneeliminates the frictional force due to O-rings. Although embodiments canuse both an O-ring and a flexible disk/membrane in different ratios, itis not possible to eliminate all O-rings with membranes since this wouldonly create an inflatable volume attached to the tube which would notproduce tension in the tube that can be transmitted to the vibratingsection of the tube between the clamps and therefore affect itsresponse.

In embodiments where two sets of O rings are used, as shown in FIG. 3,any residual frictional force can be measured and compensated for by thepressure-compensated fluid density estimator 302 using tension measuringdevices as described with reference to FIG. 2.

By tuning the force on the tube 301 F_(t), the pressure dependent termsappearing in the PDE (1) can be reduced, thereby reducing the pressuredependence of equations (2)-(6) derived from the analytical solutionthereof. Specifically, the

$\left( {{- F_{t}} + {P \cdot S_{f}}} \right)\frac{\partial^{2}\psi}{{\partial t}{\partial z}}$term is pressure dependent, where F_(t) is pressure dependent becausethe fluid 300 exerts a force on the tube 301 that depends on thepressure P of the fluid 300 (as well as other system parameters).Therefore, we want to eliminate the pressure dependence of this term:

$\begin{matrix}{\frac{\partial\left( {{- F_{t}} + {PS}_{f}} \right)}{\partial P} = 0} & (22)\end{matrix}$Referring now to the expression for F_(t) given in equation (3), and thegeometry in FIG. 3, the external force term F_(Δ) can be expressed as asum of frictional forces and force exerted by the piston 333:

$\begin{matrix}{F_{\Delta} = {{P\frac{\pi}{4}\left( {c^{2} - a^{2} + b^{2}} \right)} + F_{friction}}} & (23)\end{matrix}$Here,

$P\frac{\pi}{4}\left( {c^{2} - a^{2} + b^{2}} \right)$is the additional force exerted by the piston 333 on the tube 301quantified as the pressure P times the difference in cross-sectionalarea of the cylinder containing the piston 333 and the cross-sectionalarea of the tube 301. Note the expression for the force takes intoaccount the surfaces onto which the fluid pressure cause an axial force.The sum of these pressure contributions do not result in zero force butrather result in a pressure-dependent tension that can be tailored bythe relative sizes of these surfaces. There is therefore a variety ofgeometries which can produce the same total force on the tube. Forexample, the OD of the seal at O-rings 353/357 and 355/359 could belarger than the tube, in which case the cross-sectional area to subtractfrom

$\frac{\pi}{4}c^{2}$would be larger than the one expressed in Equation (15). For thepressures seen in downhole applications, which can be up to 30 ksi, thefrictional force is small in comparison to the main force on the tubeend due to pressure. Furthermore, its' direction, and hence sign inEquation (15), will vary and can be assumed to be zero on average.Frictional force is neglected in the design phase of thepressure-compensated densitometer, hence with F_(friction)=0 in (15) andcalculating the derivative per (14) using the full expression for F_(t)(3), Equation (14) yields:

$\begin{matrix}{{{\left( {{2v_{t}} - 1} \right)b^{2}} + {\frac{E_{t}S_{t}}{E_{c}S_{c}}\left( {c^{2} - a^{2}} \right)}} = 0} & (24)\end{matrix}$By solving for c in equation (16) using standard values for a, b, thedensitometer in FIG. 3 can be designed with the piston 333 with aprescribed diameter c such that the measured resonant frequency has aminimized dependence on pressure, as described above.

$\begin{matrix}{c = \sqrt{a^{2} + {\frac{E_{c}S_{c}}{E_{t}S_{t}}{b^{2}\left( {1 - {2v_{t}}} \right)}}}} & (25)\end{matrix}$

For example, a densitometer system can have α=0.301″, b=0.219″,A=1.800″, B=0.600″, ν_(t)=0.342 which results in c=1.010″ from equation(16) if E_(t)=E_(c). This value for c is a reasonable dimension forpractical design considerations.

Once the piston 333 is installed with diameter according to equation(16), the resonance frequency will have a much smaller dependence onpressure, due to the elimination of the

$\left( {\underset{\underset{0}{︸}}{m_{f}V^{2}} - F_{t} + {P \cdot S_{f}}} \right)\frac{\partial^{2}\psi}{\partial z^{2}}$term in Equation (1). The resonant frequency will retain a smalldependence on pressure due to the contribution of pressure on thegeometry of the vibrating tube, which affects the other terms inEquation (1). With experimentation or numerical simulations, one canfurther refine the choice of the dimension c to further reduce oreliminate the dependence of the resonance frequency on the appliedpressure.

To account for friction, the F_(Δ) term of Equation (23) can be betterdetermined using a tension measuring device as described in FIG. 2above, for example, using a full-Poisson bridge of strain gauges, usingthe following equation:

$\begin{matrix}{F_{friction} = {{F_{\Delta} - {P\frac{\pi}{4}\left( {c^{2} - a^{2} + b^{2}} \right)}} = {{\left( \frac{ɛ_{B}{E_{t}\left( {a^{2} - b^{2}} \right)}}{\left( {K_{1} - K_{2}} \right)\left( {1 - v_{t}} \right)} \right)\frac{\pi}{2}} - {P\frac{\pi}{4}\left( {c^{2} - a^{2} + {3b^{2}}} \right)}}}} & (26)\end{matrix}$In equation (26), ε_(B) is the output strain by the full-Poisson bridgecircuit with the strain gauges mounted on the tube 301.

The densitometers depicted in FIGS. 1-3 can be deployed in-line in aPWFT tester, or as a sensor in any LWD system. The measurement modules102, 202, and 302 can be configured to receive additional measurementsof formation properties including fluid bubble point, fluidcompressibility, fluid pressure, fluid temperature, fluid viscosity, andfluid thermal conductivity (i.e. changes in all of the aboveproperties).

Any of the densitometer modifications depicted by the exampledensitometers in FIGS. 1-3 can be used interchangeably. The measurementmodules can be configured to calibrate fluid density measurementsaccording to any of the theoretical characterizations given above. Forinstance, a densitometer can have dissimilar tube and clamp materials, apiston component, and tension measuring devices. Such a system can haveminimized temperature and pressure dependence and can use tensionmeasurements to compensate for residual forces acting on thedensitometer tube.

Example Densitometer Figures

FIG. 4 is an isometric view of a sealed piston at the end of adensitometer tube. A fluid enters the densitometer via a first manifoldflow path 309 and continues to a tube 301 via a port 331. The fluidexerts pressure on a piston 333 at the end of the tube 301. The piston333 is sealed to a manifold 337 with O-rings 345, 347, 349, and 351. Afluid chamber 335 situated behind the piston 333 inside of the manifold337 collects fluid leakage through the seal of the O-rings 345, 347,349, and 351. The manifold is sealed to the tube 301 via an additionalset of O-rings 353, 355, 357, and 359. The components in FIG. 4 can bepart of a densitometer system that is designed to minimize pressuredependence of density estimates of a sample fluid, for example thedensitometer in FIG. 3.

FIG. 5 is a frontal view of four strain gauges affixed to a tube in“axial full Poisson bridge” configuration. Strain gauges 501-504 areattached to the outside of a tube 500 and measure strains ε₁ 505, ε₂506, ε₃ 507, and ε₄ 508. Strain gauges 501 and 503 are aligned parallelto the axis of the tube 500 and strain gauges 502 and 504 are alignedperpendicular to the axis of the tube 500. Strain gauges 501 and 502 areon the top of the tube 500 and strain gauges 503 and 504 are on thebottom of the tube 500. This configuration of strain gauges 501-504 isdesigned to increase accuracy of axial strain measurements along thetube 500 and remove bending and temperature effects. With reference toFIG. 2, the tension measuring device 233 can comprise the strain gauges501, 502 and the tension measuring device 235 can comprise the straingauges 503, 504. The tube 500 can be any of the tubes 101, 201, or 301with reference to FIGS. 1-3. Note that even though temperature andbending effects are removed, effects due to pressure, which causes anincrease in the radial dimension, are not removed. This is the reasonfor a separate pressure measurement that is then used to calculateF_(Δ). A similar configuration of strain gauges can be used to measurestrain on the clamps 103, 203, 303 of any of the densitometers depictedin FIGS. 1-3 above (e.g., tension measuring device 237 is thecombination of strain gauge 501 and 502 and tension measuring device 239is the combination of strain gauges 503 and 504). When measuring thestrain on the clamp, there is no pressure effect but we have a largeattenuation effect due to the stiffness of the clamp compared to thetube. The axial strain on the clamp will be:

$\begin{matrix}{ɛ_{clamp} = \frac{F_{\Delta}}{{S_{c}E_{c}} + {S_{t}E_{t}}}} & (27)\end{matrix}$Here too one can use a Full Poisson bridge to determine ε_(clamp).

$\begin{matrix}{ɛ_{B} = {{\frac{\left( {K_{1} - K_{2}} \right)\left( {1 + v_{c}} \right)}{2}ɛ_{clamp}} = {\frac{(G)\left( \left( {1 + v_{c}} \right) \right.}{2\left( {1 - {v_{0}K_{T}}} \right)}ɛ_{clamp}}}} & (28)\end{matrix}$Such that

$\begin{matrix}{ɛ_{clamp} = {\frac{2ɛ_{B}}{\left( {K_{1} - K_{2}} \right)\left( {1 + v_{c}} \right)} = {2\frac{\left( {1 - {v_{0}K_{T}}} \right)ɛ_{B}}{{G\left( {1 + K_{T}} \right)}\left( {1 + v_{c}} \right)}}}} & (29)\end{matrix}$

FIGS. 6-8 refer to a densitometer assembly system in illustrativeexamples. The operations described in FIGS. 6-8 can be performed by anysystem configured to assemble, test, and calibrate a densitometer systemwith temperature, pressure, or tension compensation.

FIG. 6 is a flowchart of example operations for designing andcalibrating a cost-effective densitometer system with minimizedtemperature dependence. Various operations described in FIG. 6 can beapplied to the densitometers depicted in FIGS. 1-3 or variationsthereof. Sub-operations within each block depicted in FIG. 6 can be usedselectively or not at all.

At block 601, a densitometer assembly system designs a baselinedensitometer with identical tube and clamp materials. The densitometerassembly system chooses tube, clamp, and housing materials as well asspecification parameters. The specification parameters include innerdiameter, outer diameter of the tube and the clamp, clamp length, lengthof the tube section within the housing, length of the vibrating sectionof the tube (inside the clamp ends), etc. The inner diameter of the tubeshould be chosen such that fluid flows uninhibited through the deployeddensitometer and should depend on the volumetric flow rate of a systemin which the densitometer is deployed. Other dimensions should be chosento be cost effective, to handle loads and pressure/temperature regimesthat are typical in the system, and according to any manufacturerspecified thresholds on the materials used for the tube and clamp.

At block 603, the densitometer assembly system runs numericalsimulations with variable clamp CTEs to minimize temperature dependenceof the resonant frequency of the sample fluid in the densitometersystem. For example, equation (7) above can be used to determine thetemperature dependence of the resonant frequency based on thespecification parameters of the densitometer system chosen at block 601.

At block 605, the densitometer assembly system determines a clampmaterial that is both cost-effective and has a CTE close to the optimalCTE to minimize temperature dependence as determined at block 603.Metallic alloys can be used to achieve a desired CTE at low cost. Thedensitometer assembly system runs numerical simulations on thedensitometer system to verify that the temperature dependence of theresonant frequency is still small. Moreover, the numerical simulationscan verify that the required forces on the tube due to temperature,pressure, external forces, and forces applied to the tube and clamp whenthe densitometer is constructed do not overload the densitometer. Thesenumerical simulations can additionally determine an optimal axialtension to apply to the clamp when securing the clamp to the tube (whichwill experience an equal and opposite axial compression force). Variousfactors including the tube and clamp specification parameters, externalforces due to environmental factors (e.g., when the densitometer isdeployed in a well at significant depth), the tube and clamp materialsand the theoretical considerations for tube and clamp materials withdifferent CTEs as given by, for example, equation (7) should beconsidered.

At block 607, the densitometer assembly system determines whether thetension in the tube is within operational thresholds. This determinationcan be based on the numerical simulations at block 605. If the tensionin the tube is within operational thresholds, operations skip to block611. Otherwise, operations continue to block 609.

At block 609, the densitometer assembly system modifies the maximumoperational temperature and/or changes the clamp material so that thedensitometer system is within operational thresholds. For example, themaximum operational temperature can be lowered so that the tension inthe tube never exceeds operational limits. Alternatively, a clampmaterial can be chosen with a material that only reduces temperaturedependence of the resonant frequency of the sample fluid in thedensitometer system instead of completely eliminating the temperaturedependence. In some embodiments, the optimal axial tension to apply tothe clamp when securing the clamp to the tube can be modified so thatthe system is within operational thresholds.

At block 611, the densitometer assembly system constructs the tube andthe clamp with the prescribed specification parameters and materials.The densitometer assembly system then secures the clamp to the tube withthe optimal axial tension and affixes pressure, temperature, and tensionsensors throughout the densitometer. The pressure, temperature, andtension sensors should be placed so as to monitor these physicalquantities at locations in the densitometer that are relevant to tuningresonant frequency measurements of the sample fluid in the tube. More orless sensors can be used depending on cost, expected external factors ina deployment environment, etc.

At block 613, the densitometer assembly system tests resonant frequencyresponse of a sample fluid for various temperature, pressure, andexternal force regimes and various sample fluid types. The range oftemperature, pressure, and external forces tested should be inaccordance with likely environmental conditions of the densitometer whendeployed. In densitometer systems designed to have minimized dependenceon temperature, smaller ranges can be used for these parameters and afewer number of temperature sensors can be affixed to the densitometerat block 611.

At block 615, the densitometer assembly system calibrates amaterial-compensated fluid density estimator to compute an accuratesample fluid density estimates under various regimes tested at block613, as well as other temperature, pressure, and external force regimes.The densitometer assembly system configures the material-compensatedfluid density estimator in the densitometer to compute sample fluiddensity according to temperature, pressure, and force measurements fromthe sensors affixed at block 611 as well as calibrated values of theresonant frequency based on these measurements using the resonantfrequency response above. The densitometer can be configured to detectmultiple sample fluid types and use different pre-calibrated resonantfrequency values according to each sample fluid type.

FIG. 7 is a flowchart of example operations for designing andcalibrating a densitometer system with tension-measuring devices.Various operations described in FIG. 7 can be applied to thedensitometers depicted in FIGS. 1-3 or variations thereof.Sub-operations within each block depicted in FIG. 7 can be usedselectively or not at all.

At block 701 a densitometer assembly system determines tube and clampspecification parameters. The densitometer assembly system can determinethese parameters as described in block 601 with reference to FIG. 6. Inembodiments where a tube and a clamp of the densitometer system are madeof the same material, the tube dimensions and the clamp dimensions canbe chosen to reduce costs based on the material.

At block 703, the densitometer assembly system determines an optimalaxial tension on the clamp. The densitometer assembly system candetermine the optimal axial tension in the clamp as described in block605 with reference to FIG. 6 and further considering maximum tensionthresholds on the densitometer system when including all axial forces onthe tube. It is desirable for the vibrating tube to have a high Q, whichis an experimentally-determined value defined as the ratio of theresonance frequency (in Hertz) to the spectral distance (in Hertz)between the frequency values on either side of the resonance frequencywhere the signal is half the value of peak value when the frequency of adrive signal of constant amplitude is traversed. Typically, largertension in the tube yield higher Q values and the practicalconsiderations entering the discussion of block 605 come to limit thetension. A Q of 300 or above is considered adequate for operation oftypical densitometer.

At block 705, the densitometer assembly system constructs the tube andthe clamp with the prescribed specification parameters and materialsdetermined at blocks 701 and 703. The densitometer assembly system thensecures the clamp to the tube with the optimal axial tension computed atblock 705 and affixes tension measuring devices and other pressure andtemperature sensors throughout the densitometer. The pressure,temperature, and tension sensors should be placed so as to monitor thesephysical quantities at locations in the densitometer that are relevantto tuning resonant frequency measurements of the sample fluid in thetube. More or less sensors can be used depending on cost, expectedexternal factors in a deployment environment, etc. The temperature ofthe densitometer when the densitometer assembly system performs theoperation at block 705 should be recorded for subsequent calculations.The tension measuring devices can be affixed to opposite sides of thetube in a tube cavity that experiences vibrations and can have anoptimized configuration for measuring axial tension, for example theconfiguration in FIG. 6. Additional tension measuring devices can beplaced throughout the densitometer system to further monitor tension invarious components to ensure they do not exceed system thresholds.

At block 706, the densitometer assembly system calibrates the tensionmeasuring devices affixed to the densitometer at block 705. For example,the densitometer calibration system can apply external load cells atvarious locations of the tension measuring devices throughout thedensitometer. Measurements taken at the tension measuring devices can becalibrated against the forces measured by the load cells.

At block 707, the densitometer assembly system tests the resonantfrequency response of the sample fluid for various temperature,pressure, and external force regimes and various sample fluid types. Therange of temperature, pressure, and external forces tested should be inaccordance with likely environmental conditions of the densitometer whendeployed.

At block 709, the densitometer assembly system calibrates thetension-compensated fluid density estimator to compute accurate samplefluid density estimates under various regimes tested at block 707. Thetension-compensated fluid density estimator can estimate external axialforces on the tube using equations (14) or (21) and can use the externalaxial force estimates to more accurately estimate resonant frequency ofthe sample fluid (and therefore estimate sample fluid density) usingequation (2).

FIG. 8 is a flowchart of example operations for designing andcalibrating a densitometer system with minimized sensitivity topressure. Various operations described in FIG. 8 can be applied to thedensitometers depicted in FIGS. 1-3 or variations thereof.Sub-operations within each block depicted in FIG. 8 can be usedselectively or not at all.

At block 801 a densitometer assembly system determines tube and clampspecification parameters. The densitometer assembly system can determinethese parameters as described in block 601 with reference to FIG. 6. Inembodiments where a tube and a clamp of the densitometer system are madeof the same material, the tube dimensions and the clamp dimensions canbe chosen to reduce costs based on the material.

At block 803, the densitometer assembly system determines pistonspecification parameters that minimize pressure sensitivity of theresonant frequency of sample fluid in the densitometer. The pistonspecification parameters can be chosen so that the pressure sensitivityprescribed by solving for the c parameter in equation (25) is minimized.Additional considerations include cost of the piston material, possibledimensions and corresponding tensions of the piston for the chosenmaterial, etc.

At block 804, the densitometer assembly system runs numericalsimulations using the piston simulation parameters determined at block803. The numerical simulations can incorporate expected operationalconditions such as temperature, pressure, and external forces toestimate the net axial load on the densitometer. The axial force in thetube can be computed using equations (3) and (23) based on the pressure,piston specification parameters, and frictional force (i.e. as inequation (26)) exerted on the tube.

At block 805, the densitometer assembly system adjusts pistonspecification parameters according to the maximum pressure rating on thetube in the densitometer. If the axial load on the tube is aboveoperational thresholds, the densitometer assembly system can adjust thepiston specification parameters such as the piston diameter, the pistonmaterial, etc. Any of the previous considerations from the operations inthe previous Figures can be incorporated such as changing the tube andclamp materials/dimensions, adjust the tension with which the clamp isadhered to the tube, etc. to achieve a densitometer with minimalpressure dependence within operational thresholds.

At block 807, the densitometer assembly system constructs the tube, theclamp, and the piston with the prescribed specification parameters andmaterials determined at blocks 801 and 805. The densitometer assemblysystem then secures the clamp to the tube with the optimal axial tensioncomputed which can be computed as described in block 703 with referenceto FIG. 7 and affixes pressure, temperature, and tension sensorsthroughout the densitometer. The pressure, temperature, and tensionsensors should be placed so as to monitor these physical quantities atlocations in the densitometer that are relevant to tuning resonantfrequency measurements of the sample fluid in the tube. More or lesssensors can be used depending on cost, expected external factors in adeployment environment, etc. The temperature of the densitometer whenthe densitometer assembly system performs the operation at block 807should be recorded for subsequent calculations.

At block 809, the densitometer assembly system tests the resonantfrequency response of the sample fluid for various temperature,pressure, and external force regimes and various sample fluid types. Therange of temperature, pressure, and external forces tested should be inaccordance with likely environmental conditions of the densitometer whendeployed.

At block 811, the densitometer assembly system calibrates thepressure-compensated fluid density estimator to compute accurate samplefluid density estimates under various regimes tested at block 809. Thepressure-compensated fluid density estimator can approximate allexternal forces with this frictional force when computing the samplefluid resonant frequency from equation (3) and can calculate frictionalforces on the tube using equation (26).

The flowcharts are provided to aid in understanding the illustrationsand are not to be used to limit scope of the claims. The flowchartsdepict example operations that can vary within the scope of the claims.Additional operations may be performed; fewer operations may beperformed; the operations may be performed in parallel; and theoperations may be performed in a different order. For example, theoperations depicted in blocks 601, 603 can be performed in parallel orconcurrently. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented byprogram code. The program code may be provided to a processor of ageneral-purpose computer, special purpose computer, or otherprogrammable machine or apparatus.

As will be appreciated, aspects of the disclosure may be embodied as asystem, method or program code/instructions stored in one or moremachine-readable media. Accordingly, aspects may take the form ofhardware, software (including firmware, resident software, micro-code,etc.), or a combination of software and hardware aspects that may allgenerally be referred to herein as a “circuit,” “module” or “system.”The functionality presented as individual modules/units in the exampleillustrations can be organized differently in accordance with any one ofplatform (operating system and/or hardware), application ecosystem,interfaces, programmer preferences, programming language, administratorpreferences, etc.

Any combination of one or more machine-readable medium(s) may beutilized. The machine-readable medium may be a machine-readable signalmedium or a machine-readable storage medium. A machine-readable storagemedium may be, for example, but not limited to, a system, apparatus, ordevice, that employs any one of or combination of electronic, magnetic,optical, electromagnetic, infrared, or semiconductor technology to storeprogram code. More specific examples (a non-exhaustive list) of themachine-readable storage medium would include the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a portable compact disc read-only memory (CD-ROM), anoptical storage device, a magnetic storage device, or any suitablecombination of the foregoing. In the context of this document, amachine-readable storage medium may be any tangible medium that cancontain or store a program for use by or in connection with aninstruction execution system, apparatus, or device. A machine-readablestorage medium is not a machine-readable signal medium.

A machine-readable signal medium may include a propagated data signalwith machine-readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Amachine-readable signal medium may be any machine-readable medium thatis not a machine-readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device.

Program code embodied on a machine-readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of thedisclosure may be written in any combination of one or more programminglanguages, including an object oriented programming language such as theJava® programming language, C++ or the like; a dynamic programminglanguage such as Python; a scripting language such as Perl programminglanguage or PowerShell script language; and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may execute entirely on astand-alone machine, may execute in a distributed manner across multiplemachines, and may execute on one machine while providing results and oraccepting input on another machine.

The program code/instructions may also be stored in a machine-readablemedium that can direct a machine to function in a particular manner,such that the instructions stored in the machine-readable medium producean article of manufacture including instructions which implement thefunction/act specified in the flowchart and/or block diagram block orblocks.

FIG. 9 depicts an example computer system with a densitometer fluiddensity estimator. The computer system includes a processor unit 901.The computer system includes memory 907. The memory 907 may be systemmemory or any one or more of the above already described possiblerealizations of machine-readable media. The computer system alsoincludes a bus 903 and a network interface 905. The system also includesa densitometer fluid density estimator 911. The densitometer fluiddensity estimator 911 can accurately determine fluid density of fluidinside a densitometer using pressure, temperature, and forcemeasurements such that the temperature dependence and pressuredependence of the densitometer response is minimized and external axialforces on a tube in the densitometer are accurately measurement andcompensated for by the densitometer fluid density estimator 911. Any oneof the previously described functionalities may be partially (orentirely) implemented in hardware and/or on the processor unit 901. Forexample, the functionality may be implemented with an applicationspecific integrated circuit, in logic implemented in the processor unit901, in a co-processor on a peripheral device or card, etc. Further,realizations may include fewer or additional components not illustratedin FIG. 9 (e.g., video cards, audio cards, additional networkinterfaces, peripheral devices, etc.). The processor unit 901 and thenetwork interface 905 are coupled to the bus 903. Although illustratedas being coupled to the bus 903, the memory 907 may be coupled to theprocessor unit 901.

While the aspects of the disclosure are described with reference tovarious implementations and exploitations, it will be understood thatthese aspects are illustrative and that the scope of the claims is notlimited to them. In general, techniques for fluid density measurementsfrom a densitometer with improved accuracy as described herein may beimplemented with facilities consistent with any hardware system orhardware systems. Many variations, modifications, additions, andimprovements are possible.

Example Drilling Application

FIG. 10 is a schematic diagram of a drilling rig system with adensitometer. For example, in FIG. 10 it can be seen how a system 1064may also form a portion of a drilling rig 1002 located at the surface1004 of a well 1006. Drilling of oil and gas wells is commonly carriedout using a string of drill pipes connected together so as to form adrilling string 1008 that is lowered through a rotary table 1010 into awellbore or borehole 1012. Here a drilling platform 1086 is equippedwith a derrick 1088 that supports a hoist.

The drilling rig 1002 may thus provide support for the drill string1008. The drill string 1008 may operate to penetrate the rotary table1010 for drilling the borehole 1012 through subsurface formations 1014.The drill string 1008 may include a kelly 1016, drill pipe 1018, and abottom hole assembly 1020, perhaps located at the lower portion of thedrill pipe 1018.

The bottom hole assembly 1020 may include drill collars 1022, a downhole tool 1024, and a drill bit 1026. The drill bit 1026 may operate tocreate a borehole 1012 by penetrating the surface 1004 and subsurfaceformations 1014. The down hole tool 1024 may comprise any of a number ofdifferent types of tools including MWD tools, LWD tools, and others.

During drilling operations, the drill string 1008 (perhaps including thekelly 1016, the drill pipe 1018, and the bottom hole assembly 1020) maybe rotated by the rotary table 1010. In addition to, or alternatively,the bottom hole assembly 1020 may also be rotated by a motor (e.g., amud motor) that is located down hole. The drill collars 1022 may be usedto add weight to the drill bit 1026. The drill collars 1022 may alsooperate to stiffen the bottom hole assembly 1020, allowing the bottomhole assembly 1020 to transfer the added weight to the drill bit 1026,and in turn, to assist the drill bit 1026 in penetrating the surface1004 and subsurface formations 1014.

During drilling operations, a mud pump 1032 may pump drilling fluid(sometimes known by those of ordinary skill in the art as “drillingmud”) from a mud pit 1034 through a hose 1036 into the drill pipe 1018and down to the drill bit 1026. The drilling fluid can flow out from thedrill bit 1026 and be returned to the surface 1004 through an annulararea 1040 between the drill pipe 1018 and the sides of the borehole1012. The drilling fluid may then be returned to the mud pit 1034, wheresuch fluid is filtered. In some embodiments, the drilling fluid can beused to cool the drill bit 1026, as well as to provide lubrication forthe drill bit 1026 during drilling operations. Additionally, thedrilling fluid may be used to remove subsurface formation 1014 cuttingscreated by operating the drill bit 1026. The drill pipe furthercomprises a densitometer 1017 configured to receive sample fluid,accurately compute sample fluid density. The densitometer 1017 iscommunicatively coupled to a logging system 1096 and sends the computedsample fluid density to the logging system 1096. The densitometer 1017can have minimized temperature and pressure dependence in its' computedsample fluid density and can compensate for external forces usingtension measuring devices as described above.

Example Wireline Application

FIG. 11 depicts a schematic diagram of a wireline system with adensitometer. A system 1100 can be used in an illustrative loggingenvironment with a drillstring removed, in accordance with someembodiments of the present disclosure.

Subterranean operations may be conducted using a wireline system 1120once the drillstring has been removed, though, at times, some or all ofthe drillstring may remain in a borehole 1114 during logging with thewireline system 1120. The wireline system 1120 may include one or morelogging tools 1126 that may be suspended in the borehole 1114 by aconveyance 1115 (e.g., a cable, slickline, or coiled tubing). Thelogging tool 1126 may be communicatively coupled to the conveyance 1115.The conveyance 1115 may contain conductors for transporting power to thewireline system 1120 and telemetry from the logging tool 1126 to alogging facility 1144. Alternatively, the conveyance 1115 may lack aconductor, as is often the case using slickline or coiled tubing, andthe wireline system 1120 may contain a control unit 1134 that containsmemory, one or more batteries, and/or one or more processors forperforming operations and storing measurements. The logging tool 1126further comprises a densitometer 1117 configured to receive samplefluid, accurately compute sample fluid density, and forward the computedsample fluid density to the logging facility 1144. The densitometer 1117can have minimized temperature and pressure dependence in its' computedsample fluid density and can compensate for external forces usingtension measuring devices as described above.

In certain embodiments, the control unit 1134 can be positioned at thesurface, in the borehole (e.g., in the conveyance 1115 and/or as part ofthe logging tool 1126) or both (e.g., a portion of the processing mayoccur downhole and a portion may occur at the surface). The control unit1134 may include a control system or a control algorithm. In certainembodiments, a control system, an algorithm, or a set ofmachine-readable instructions may cause the control unit 1134 togenerate and provide an input signal to one or more elements of thelogging tool 1126, such as the sensors along the logging tool 1126. Theinput signal may cause the sensors to be active or to output signalsindicative of sensed properties. The logging facility 1144 (shown inFIG. 11 as a truck, although it may be any other structure) may collectmeasurements from the logging tool 1126, and may include computingfacilities for controlling, processing, or storing the measurementsgathered by the logging tool 1126. The computing facilities may becommunicatively coupled to the logging tool 1126 by way of theconveyance 1115 and may operate similarly to the control unit 1134. Incertain example embodiments, the control unit 1134, which may be locatedin logging tool 1126, may perform one or more functions of the computingfacility.

The logging tool 1126 includes a mandrel and a number of extendible armscoupled to the mandrel. One or more pads are coupled to each of theextendible arms. Each of the pads have a surface facing radially outwardfrom the mandrel. Additionally, at least sensor disposed on the surfaceof each pad. During operation, the extendible arms are extended outwardsto a wall of the borehole to extend the surface of the pads outwardagainst the wall of the borehole. The sensors of the pads of eachextendible arm can detect image data to create captured images of theformation surrounding the borehole.

Plural instances may be provided for components, operations orstructures described herein as a single instance. Finally, boundariesbetween various components, operations and data stores are somewhatarbitrary, and particular operations are illustrated in the context ofspecific illustrative configurations. Other allocations of functionalityare envisioned and may fall within the scope of the disclosure. Ingeneral, structures and functionality presented as separate componentsin the example configurations may be implemented as a combined structureor component. Similarly, structures and functionality presented as asingle component may be implemented as separate components. These andother variations, modifications, additions, and improvements may fallwithin the scope of the disclosure.

Example Densitometer Graphs

FIGS. 12-19 depict example graphs of various measured and intrinsicsystem and sample fluid properties for densitometer systems. Thesegraphs are intended for illustrative purposes and a real-worlddensitometer system should additionally compensate for operationalconditions such as expected temperature, pressure, and external forces,orientation of the tool, sampling rate, sensor locations, etc. Exceptwhere noted otherwise, the pressure inside the densitometer is assumedto be 0 psig in the simulations and a fluid density of 1.000 g/cm³ isassumed in all cases.

FIG. 12 depicts a graph 1200 of resonant frequency of a sample fluid ina densitometer versus temperature for a standard design of densitometerwhere the tube and housing of the densitometer are made of titaniumalloy (Ti-6AI-4V, grade 5) with a CTE of 9.20×10⁻⁶° C. To illustrate thedesign options available, a “CTEfactor” is used as multiplication factorfor the value of clamp CTE material relative to the CTE of the tubematerial. Different plots are shown for CTEfactor=0.5, 1.0, 1.5793,1.7499, and 2.0. A CTEfactor of 1.0 represents the standard design forwhich clamp and tube have the same CTE. It can be seen in FIG. 11 thatfor CTEfactor=1.0, the resonance frequency changes with temperature.With CTEfactor=1.7499, corresponding to a clamp CTE of 16.10×10⁻⁶/° C.,the temperature dependence of the resonant frequency is nearlyeliminated and represents optimal ratio of CTE to minimize thedependence of the densitometer to temperature. This informs theselection of the clamp material. In some embodiments there is not a realmaterial that is a perfect match for this CTE. In this case, the metalInconel 706 has an average CTE of 14.53×10⁻⁶/° C. in the range 26-200°C. and is close to the desired value, as well as being suitable for thisapplication. This corresponds to the CTEfactor value of 1.5793 alsographed in FIG. 12.

FIG. 13 depicts a graph 1300 of axial force on the vibrating tube versustemperature. For a CTE of 1.7499, at a deployed temperature of 200° C.,the densitometer tube receives a

${\left( {16.10 - 9.20} \right) \times \frac{\mu ɛ}{{^\circ}\mspace{14mu}{C.}} \times \left( {200 - 25} \right){^\circ}\mspace{14mu}{C.}} = {1208\mspace{14mu}{\mu\epsilon}}$temperature-induced axial strain, which may be above the upper limit oftolerable strain for such a densitometer. For a CTEfactor of 1.5793,applicable to the use of Inconel 706 for the clamp, mere is a

${\left( {14.53 - 9.20} \right) \times \frac{\mu ɛ}{{^\circ}\mspace{14mu}{C.}} \times \left( {200 - 25} \right){^\circ}\mspace{14mu}{C.}} = {933\mspace{14mu}{\mu\epsilon}}$temperature-induced axial strain, which is less severe. With thisconsideration, the upper limit of temperature specification for the toolcan be reduced, or a clamp material with an even lower CTEfactor can bechosen such that the temperature-dependence of the densitometer responseis not optimal but still smaller than when the same material is used forthe clamp as for the tube. Hence, a design can be selected that takesinto account both temperature-dependence, and allowable range oftemperature.

FIG. 14 depicts a graph 1400 of resonant frequency versus pressure for astandard densitometer design, without the piston of FIGS. 3 and 4, andnot optimized for reduced pressure sensitivity. It can be seen thatincreasing the pressure results in a decrease in the resonant frequencyof the vibrating tube.

FIG. 15 depicts a graph 1500 of resonant frequency versus pressure for apressure-compensated densitometer with a predetermined piston diameter.The predetermined piston diameter is the value c=1.06507 inches isdetermined using equation (17). Although pressure dependence is reducedin graph 1300, it is not eliminated. As explained in the description ofFIG. 3, this is because Equation (17) only covers the P-dependencebrought by the second term of Equation 1, which is dominant but notalone in determining the P-dependence of the frequency response of thevibrating tube. Varying the value of c near the calculated andcalculating the slope of the response (slope of the curves such as thosein FIG. 14), establishing a relationship between the slope value and thevalue of c, and then finding, by standard linear regression (or otherstandard optimization technique) the value of c that minimizes theslope, we find that c=1.003311 inches, in the illustrated case, providesthe optimal value to eliminate the pressure-dependence of the design.

FIG. 16 depicts a graph 1600 of pressure versus resonant frequency for apressure-compensated densitometer with an optimized piston diameter.Varying the value of c calculated with reference to FIG. 15 near thecalculated value c=1.06507 inches, and calculating the slope of theresponse (slope of the curves in FIG. 15), establishing a relationshipbetween the slope value and the value of c, and then finding by standardlinear regression (or other standard optimization technique) the valueof c that minimizes the slope, the optimal value is c=1.003311 inchesand the results are plotted in FIG. 16. The pressure dependence of thedensitometer depicted by graph 1600 is nearly eliminated.

FIG. 17 depicts a graph 1700 of net axial force on the tube versuspressure for the densitometer design in FIG. 14 (T=25° C. case). FIG. 15shows the piston has a diameter (i.e. the tube end outer diameter) suchthat pressure dependence is negligible. Conversely, FIG. 16 shows thatthe net axial force on the tube quickly increases with increasedpressure. Using 4000 lbf as a typical design threshold, we see that thisis exceeded when P reaches about 5000 psi. Such large tensions areneeded in this case because this tension is acting on the sections oftube external to the clamp, whereas it is the tension of the tubesection (of length L) between the two clamp ends that affect theresonance frequency of the tube. Only a small fraction of the externalforce is transferred to the vibrating tube, and this is due to therigidity of the clamp body compared to that of the tube incross-sections in the region of length L between the clamps. This axialforce is a serious design consideration for densitometer systems. Inthis case it would limit the pressure rating to 5000 psi, which is toosmall for most downhole applications, where a pressure rating of 20000psi or above is typically needed.

FIG. 18 depicts a graph 1800 of resonant frequency versus pressure for arevised pressure compensated densitometer. Here the densitometer has atube with an outer diameter 0.301″, an inner diameter 0.220″, a tubematerial Ti-6AI-4V grade 5 alloy, the same as for the design in FIGS. 15and 16. However, the clamp OD is now decreased reduced from 1.800″ to1.000″. In this case, the optimal value of c=0.550896 inches isdetermined using the same procedure with reference to FIGS. 15 and 16.FIG. 18 shows the tube resonance frequency vs. pressure for these newdimensions and we see that the response is flat with pressure for bothT=25° C. and T=100° C.

FIG. 19 depicts a graph 1900 of net axial force on the tube versuspressure for the densitometer design in FIG. 18. The graph 1900 showsthat the change to a smaller OD for the clamp results in a much smallerforce on the external tube being needed to cause the samepressure-dependence-removing axial force on the vibrating tube. Theresulting design has a pressure rating of 20000 psi.

As used herein, the term “or” is inclusive unless otherwise explicitlynoted. Thus, the phrase “at least one of A, B, or C” is satisfied by anyelement from the set {A, B, C} or any combination thereof, includingmultiples of any element.

What is claimed is:
 1. A method comprising: inducing, with a vibrationsource attached to a tube of a densitometer, a vibration in the tubewhich contains a fluid; detecting, from a vibration detector attached tothe tube, an indication of the vibration in the tube; and determining anestimate ρ_(f) of a fluid density of the fluid in the tube based, atleast in part, on the vibration indication, a plurality of measurementsreceived from a one or more sensors attached to the densitometer, andproperties of a piston attached to an end of the tube of thedensitometer.
 2. The method of claim 1, further comprising attaching thepiston to the end of the tube of the densitometer.
 3. The method ofclaim 1, wherein the estimate ρ_(f) of the fluid density of the fluid inthe tube is determined in accordance with${f_{0} = {\frac{\beta_{0}^{2}}{2\pi\; L^{2}}\sqrt{\frac{E_{t}I}{m_{t} + {\rho_{f}S_{f}}}}}},$wherein f₀ is a resonant frequency of the vibration in the tube, β₀ is acoefficient that depends on the plurality of measurements and materialproperties of the densitometer, E_(t) is a Young's modulus of a materialof the tube, m_(t) is a mass per unit length of the tube, S_(f) is across sectional area of the fluid in the tube, I is an area moment ofinertia of the tube, and L is a length of the tube.
 4. The method ofclaim 1, wherein determining the estimate ρ_(f) of the fluid density ofthe fluid in the tube comprises determining estimate ρ_(f) of the fluiddensity of the fluid in the tube based, at least in part, on a resonantfrequency obtained with the tube containing predetermined fluid sampleswith predetermined fluid density values at a controlled temperaturevalue for the densitometer, a controlled pressure value for the fluid,and a controlled or measured external force value on the tube.
 5. Themethod of claim 1, wherein the plurality of measurements comprises atleast one of temperature measurements, pressure measurements, tensionmeasurements, or force measurements.
 6. The method of claim 1, furthercomprising determining a diameter c of the piston to reducepressure-dependence of the estimate ρ_(f) of the fluid density of thefluid in the tube.
 7. The method of claim 6, wherein the diameter c ofthe piston is determined in accordance with${c = \sqrt{a^{2} + {\frac{E_{c}S_{c}}{E_{t}S_{t}}{b^{2}\left( {1 - {2v_{t}}} \right)}}}},$wherein α is an outer diameter of the tube, b is an inner diameter ofthe tube, E_(c) is a coefficient of thermal expansion (CTE) of a clampof the densitometer, S_(c) is a cross-sectional area of the clamp, E_(t)is a CTE of the tube, S_(t) is a cross-sectional area of the tube, andρ_(t) is a Poisson's ratio of the tube.
 8. The method of claim 6,wherein the diameter c of the piston is chosen such that thedensitometer is within operational thresholds for environmentaltemperature, environmental pressure, and environmental force.
 9. Anon-transitory machine-readable medium having program code forestimating fluid density of a fluid, the program code comprising programcode to: control a vibration source to induce a vibration in a tube of adensitometer; receive an indication of vibration detected by a vibrationdetector attached to the tube at a different location than the vibrationsource; calculate an estimate ρ_(f) of the fluid density of the fluid inthe tube, wherein the program code to calculate the estimate of thefluid density comprises program code to, input the indication of thevibration, a plurality of measurements received from one or more sensorsattached to the densitometer, and properties of a piston attached to anend of the tube into a calibrated fluid density model; and generate,from the calibrated fluid density model, the estimate ρ_(f) of the fluiddensity of the fluid in the tube.
 10. The non-transitorymachine-readable medium of claim 9, wherein the program code togenerate, from the calibrated fluid density model, the estimate ρ_(f) ofthe fluid density of the fluid in the tube comprises program code togenerate, from the calibrated fluid density model, the estimate ρ_(f) ofthe fluid density of the fluid in the tube in accordance with${f_{0} = {\frac{\beta_{0}^{2}}{2\pi\; L^{2}}\sqrt{\frac{E_{t}I}{m_{t} + {\rho_{f}S_{f}}}}}},$wherein f₀ is a resonant frequency of the vibration in the tube, β₀ is acoefficient that depends on first material properties, second materialproperties, and the plurality of measurements, E_(t) is a Young'smodulus of the first material, m_(t) is a mass per unit length of thetube, S_(f) is a cross sectional area of the fluid in the tube, I is anarea moment of inertia of the tube, and L is a length of the tube. 11.The non-transitory machine-readable medium of claim 9, wherein theplurality of measurements comprises at least one of temperaturemeasurements, pressure measurements, force measurements, or tensionmeasurements.
 12. The non-transitory machine-readable medium of claim 9,further comprising program code to determine a diameter c of the pistonto reduce force-dependence of the estimate ρ_(f) of the fluid density ofthe fluid in the tube.
 13. The non-transitory machine-readable medium ofclaim 12, wherein the program code to determine the diameter c of thepiston comprises program code to determine the diameter c of the pistonin accordance with${c = \sqrt{a^{2} + {\frac{E_{c}S_{c}}{E_{t}S_{t}}{b^{2}\left( {1 - {2v_{t}}} \right)}}}},$wherein α is an outer diameter of the tube, b is an inner diameter ofthe tube, E_(c) is a coefficient of thermal expansion (CTE) of a clampof the densitometer, S_(c) is a cross-sectional area of the clamp, E_(t)is a CTE of the tube, S_(t) is a cross-sectional area of the tube, andρ_(t) is a Poisson's ratio of the tube.
 14. An apparatus comprising: atube that receives a fluid; a clamp encompassing the tube; a vibrationsource attached to the tube; a vibration detector attached to the tube;one or more sensors attached to the apparatus; a piston attached to anend of the tube; and a computing device coupled to control the vibrationsource to induce a vibration in the tube and coupled to the vibrationdetector to receive an indication of the vibration in the tube detectedby the vibration detector, the computing device programmed to determinean estimate ρ_(f) of a fluid density of the fluid in the tube based, atleast in part, on a plurality of measurements received from the one ormore sensors attached to the apparatus and properties of a pistonattached to an end of the tube.
 15. The apparatus of claim 14, whereinthe computing device comprises a machine-readable medium having storedtherein program code to determine the estimate ρ_(f) of the fluiddensity of the fluid in the tube in accordance with${f_{0} = {\frac{\beta_{0}^{2}}{2\pi\; L^{2}}\sqrt{\frac{E_{t}I}{m_{t} + {\rho_{f}S_{f}}}}}},$wherein f₀ is a resonant frequency of the vibration in the tube, β₀ is acoefficient that depends on first material properties, second materialproperties, and the plurality of measurements, E_(t) is a Young'smodulus of the first material, m_(t) is a mass per unit length of thetube, S_(f) is a cross sectional area of the fluid in the tube, I is anarea moment of inertia of the tube, and L is a length of the tube. 16.The apparatus of claim 14, wherein the computing device is calibrated todetermine the estimate ρ_(f) of the fluid density of the fluid in thetube based, at least in part, on a resonant frequency obtained with thetube containing predetermined fluid samples with predetermined fluiddensity values at a controlled temperature value for the densitometer, acontrolled pressure value for the fluid, and a controlled diameter c ofthe piston.
 17. The apparatus of claim 14, wherein the plurality ofmeasurements comprises at least one of temperature measurements,pressure measurements, tension measurements, and force measurements. 18.The apparatus of claim 14, wherein the computing device comprises amachine-readable medium having stored therein program code to determinea diameter c of the piston to reduce force-dependence of the estimateρ_(f) of the fluid density of the fluid in the tube.
 19. The apparatusof claim 18, wherein the program code to determine the diameter c of thepiston is comprises the program code to determine the diameter c of thepiston in accordance with${c = \sqrt{a^{2} + {\frac{E_{c}S_{c}}{E_{t}S_{t}}{b^{2}\left( {1 - {2v_{t}}} \right)}}}},$wherein α is an outer diameter of the tube, b is an inner diameter ofthe tube, E_(c) is a coefficient of thermal expansion (CTE) of a clampof the densitometer, S_(c) is a cross-sectional area of the clamp, E_(t)is a CTE of the tube, S_(t) is a cross-sectional area of the tube, andρ_(t) is a Poisson's ratio of the tube.
 20. The apparatus of claim 18,wherein the program code to determine the diameter c of the pistoncomprises the program code to determine the diameter c of the pistonsuch that the densitometer is within operational thresholds forenvironmental temperature, environmental pressure, and environmentalforce.